I see this in theory, but I'm having a hard time thinking of geometric problems that violate intuition and that are appropriate to teach people who have no calc

I see this in theory, but I'm having a hard time thinking of geometric problems that violate intuition and that are appropriate to teach people who have no calculus and limited algebraic skills (admittedly this limits statistical problems too). I'm actually curious if you have any good examples, because I don't. And I think that's where my objection comes in: if you spend the entire first year just developing that intuition but forcing a rote procedure, you'll have lost the majority of people already. Proof isn't arithmetic and we shouldn't teach it in the same way. Sure develop the process on the easy method first, but then show people why it's important to not skip steps.